Enterprise modeling is the process of characterizing a real-world enterprise, using mathematical representations, graphs, and/or pictures. An enterprise planning model often utilizes mathematical modeling techniques to analyze complex real-world scenarios, typically with the goal of improving or optimizing performance. Accordingly, an enterprise planning model can ideally provide insight into past, current, and future operating performance, enabling managers to spot trends, identify opportunities, and affect outcomes.
Multi-dimensional optimization entails determining a set of values that maximizes (or minimizes) a function of many decision variables. The types of mathematical relationships (for example, linear, nonlinear, or discontinuous) between the objective, the constraints, and the decision variables determine how difficult the optimization is to solve. The types of mathematical relationships also determine the solution methods or algorithms that can be used for optimization and the confidence that the solution is truly optimal. Such multi-dimensional optimization is problematic in that there is no known single multi-dimensional optimization strategy that can tackle all problems in a satisfactory way. In addition, the presence of constraints, even of simple ones, enhances this difficulty.
One known strategy for performing multi-dimensional optimization is an exhaustive search method in which an entire configuration space of scenarios (i.e., possible combinations of the variables) is performed to select an optimum out of all possibilities. The exhaustive search method will yield the global optimum. Unfortunately, this method is extremely computationally slow and therefore not applicable in practical situations.
Other strategies entail the iterative gradient search methods. These methods use information of the first and possibly second order derivatives of the criterion function to derive optimal search directions towards the optimum. Gradient search methods guarantee decreasing criterion values in successive iterations. The gradient search methods improve significantly over the exhaustive search method, but are still computationally slow and costly. A further disadvantage of these methods is that they are sensitive to the initial estimates of the unknowns if the criterion function has more than one optimum. As such, the gradient search algorithm may converge to a local optimum instead of the desired global optimum. A global optimum is one in which are no other feasible solutions with better objective function values. In contrast, a local optimum is one in which there are no other feasible solutions “in the vicinity” with better objective function values.
Much attention has been directed toward developing algorithms that circumvent convergence toward a local optimum. Two such algorithms are simulated annealing and genetic algorithms. The simulated annealing technique is essentially a local search, in which a move to an inferior solution is allowed with a probability that decreases as the process progresses. As such, there will always be a chance that a solution with a less good value might be retained in preference to a better solution. Thus, fine tuning of parameter settings is required.
Genetic algorithms are search techniques based on an abstract model of Darwinian evolution. Solutions are represented by fixed length strings over some alphabet (“gene” alphabet). Each string can be thought of as a “chromosome”. The value of the solution represents the fitness of the chromosome. Survival of the fittest principle is then applied to create a new generation with slow increase of average fitness. Accordingly, genetic algorithms also have the facility of allowing some weak members to survive in the solution pool, but typically have mechanisms for favoring fitter solutions.
Both simulated annealing and genetic algorithms have a fair chance of circumventing convergence toward a local optimum. In addition, both methods are much faster than the exhaustive search method. However, both methods are still computationally expensive as compared with gradient search methods.
Another problem with conventional optimization algorithms arises when optimizing in the presence of coupled decision variables. As used herein, the term “coupled” refers to decision variables and other objectives within an enterprise planning model that are connected causally to influence one another. This coupling further complicates optimization problems, leading to even more computational expense.
Thus, what is needed is a technique, within an enterprise planning model, that can efficiently and cost effectively solve complex optimization problems.